Point-biserial Correlations and Item Fits

The relation between point-biserial correlation discrimination estimates (rpbis) and Rasch fit statistics (RFS) is nearly monotonic apart from the effect of item-person targeting on point-biserial ceilings.

Maximal point-biserial or point-measure correlation is (Normal Ordinate)/square-root(P-Value * (1 - P-Value)) Figure corrected on 7/25/2011

The Figure shows the maximum possible point-biserial correlations with complete dichotomous data for items with different p-values relative to a normally-distributed sample of person abilities.

a. When item response residuals are noisy, disturbed by unexpected variation, they contradict the item's calibration. This makes the item rpbis smaller than typical of these data and the item RFS larger than expected statistically.

b. When item response residuals are muted, subdued by unexpected lack of variation, they indicate a contraction from the modelled level of independence among residuals and hence an underestimation of standard errors of measurement based on this model. This makes the item rpbis larger than typical of these data and the item RFS smaller than expected statistically.

There is an important difference in the utility of these alternative fit statistics. For the rpbis based on raw scores, the most we can say is "smaller (or larger) than we are used to". We don't know where the value we are observing is placed in the possible range. We don't know whether that value is acceptable, undesirably large or undesirably small. The rpbis is a misfit statistic but of unknown size and significance. All we know for certain, (and this is useful in detecting miscoded data), is that negative rpbis means that the observed responses to that item contradict the general meaning of the test.

For RFS based on a measurement system, we can say - "larger (or smaller) than expected statistically". The basis for expectation is a statistical model for a standard distribution of residuals. RFS give a size and a significance to misfit. The size enables us to identify misfit big enough to disturb measurement. The significance indicates what proportion of all possible misfit statistics would be better fitting than this one.

One author writes: "Ideally, it is recommended that items have point-biserials ranging from 0.30 to 0.70 (Allen, M. J. and Yen, W. M. (1979). Introduction to Measurement Theory. Waveland Press, Inc. Prospect Heights Il)".
A rule such as this cuts off the very easy and very hard items, and may even eliminate good-fitting on-target items.

Point-biserials and item fits. Wright BD. … Rasch Measurement Transactions, 1992, 5:4 p.174

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Rasch Measurement Transactions & Rasch Measurement research papers - free	An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse	Rasch Measurement Theory Analysis in R, Wind, Hua	Applying the Rasch Model in Social Sciences Using R, Lamprianou	El modelo métrico de Rasch: Fundamentación, implementación e interpretación de la medida en ciencias sociales (Spanish Edition), Manuel González-Montesinos M.
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar	Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch	Rasch Models for Measurement, David Andrich	Constructing Measures, Mark Wilson	Best Test Design - free, Wright & Stone Rating Scale Analysis - free, Wright & Masters
Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias	Diseño de Mejores Pruebas - free, Spanish Best Test Design	A Course in Rasch Measurement Theory, Andrich, Marais	Rasch Models in Health, Christensen, Kreiner, Mesba	Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen
Rasch Books and Publications: Winsteps and Facets
Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene	Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver	Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone	Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale	Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes	Rasch Models for Solving Measurement Problems (Facets), George Engelhard, Jr. & Jue Wang	Statistical Analyses for Language Testers (Facets), Rita Green	Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind	Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind	Rasch Measurement: Applications, Khine	Winsteps Tutorials - free Facets Tutorials - free	Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre	Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan

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